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EIGENVALUES AND EIGENVECTORS

ELECTRONIC VERSION OF LECTURE

May 16, 2023

OUTLINE

1 EIGENVECTOR-EIGENVECTOR OF A MATRIX

OUTLINE

1 EIGENVECTOR-EIGENVECTOR OF A MATRIX

2 DIAGONALIZATION-APPLICATION

3 MATLAB

DEFINITION 1.

If A is an n × n matrix, then a nonzero vector X ∈ R n^ , X ̸= 0 is called an eigenvector of A if AX = λ. X for some scalar λ.

DEFINITION 1.

If A is an n × n matrix, then a nonzero vector X ∈ R n^ , X ̸= 0 is called an eigenvector of A if AX = λ. X for some scalar λ. The scalar λ is called an eigenvalue of A and X is said to be an eigenvector corresponding to λ.

If λ is an eigenvalue of A ⇔ ∃ X ̸= 0 : AX = λ. X ⇔ AX − λX = 0 ⇔ ( A − λI ). X = 0.

If λ is an eigenvalue of A ⇔ ∃ X ̸= 0 : AX = λ. X ⇔ AX − λX = 0 ⇔ ( A − λI ). X = 0. This homogeneous linear system has non-zero solution X ̸= 0, thus

det ( A − λI ) = 0.

DEFINITION 1.

If A is an n × n matrix, then λ is an eigenvalue of A if and only if χA ( λ ) = det ( A − λI ) = 0. This is called the characteristic equation of A. The polynomial χA ( λ ) = det ( A − λI ) is called the characteristic polynomial.

FINDING EIGENVALUES AND EIGENVECTORS

Let A be an n × n matrix An eigenvalue of A is a scalar λ such that: det ( A − λI ) = 0

EXAMPLE 1.

Find eigenvalues and eigenvectors of A =

μ 1 0 0 − 1

EXAMPLE 1.

Find eigenvalues and eigenvectors of A =

μ 1 0 0 − 1

Solution.

Solve the equation

1 − λ 0 0 − 1 − λ

¯ =^ λ

(^2) − 1 = 0 ⇒ the matrix has

2 eigenvalues λ = 1, λ = −1. Find eigenvectors corresponding to λ = 1 ⇒ Solve the homogeneous system

μ 0 0 0 − 2

X = 0, the general solution of this

system is X =

μ α 0

, α ∈ R: eigenvectors associated to λ = 1.

With λ = − 1 ⇒ Solve:

μ 2 0 0 0

X = 0 ⇒ X =

μ 0 α

α ∈ R: eigenvectors associated to λ = −1.

EXAMPLE 1.

Find eigenvalues and eigenvectors of A =

μ 1 2 − 2 1

Solution

Solve | A − λI | = 0 ⇒ λ = 1 + 2 i , λ = 1 − 2 i.

EXAMPLE 1.

Find eigenvalues and eigenvectors of A =

μ 1 2 − 2 1

Solution

Solve | A − λI | = 0 ⇒ λ = 1 + 2 i , λ = 1 − 2 i.

With λ = 1 + 2 i we solve

μ − 2 i 2 − 2 − 2 i

X = 0 ⇒ X = α

μ − i 1

, α ∈ R.

EXAMPLE 1.

Find the eigenvalues and corresponding eigenvectors of

A =

Solution

Solve | A − λI | = 0 ⇒ λ = 2.

EXAMPLE 1.

Find the eigenvalues and corresponding eigenvectors of

A =

Solution

Solve | A − λI | = 0 ⇒ λ = 2.

With λ = 2, find the eigenvectors by solving

 X = 0